 Solution

ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2004/section-a/2/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Treat

α

as a constant and use the canonical ensemble identity

F = - k_{B} T ln Z

. The given formal partition function gives

F = α k_{B} T^{3} N^{2} / V

. The pressure is its negative volume derivative at fixed

T, N

P = - (\frac{\partial F}{\partial V})_{T, N} = α k_{B} T^{3} \frac{N ^{2}}{V ^{2}}, P V^{2} = α k_{B} T^{3} N^{2} .

Equivalently

P = k_{B} T (\partial ln Z / \partial V)_{T, N}

. The explicit temperature dependence in

ln Z

must be retained; this is not the ideal-gas pressure law.

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